This paper proposes a new way to build investment portfolios that looks at how assets move together across different time horizons and for different sizes of price moves. The authors replace the usual variance-based risk measure with a signed fluctuation function from multifractal cross-correlation analysis (MFCCA). That function is indexed by a time scale s (the investment horizon) and a fluctuation order q (which puts more weight on large or small moves). Because MFCCA keeps the sign of local co-movements, assets that tend to move in opposite directions can reduce measured portfolio risk, while assets that move together increase it. For the special case q = 2 the measure reduces to a scale-dependent version of the familiar mean–variance rule from Modern Portfolio Theory (MPT).

The researchers build a “mean–MFCCA” portfolio model that embeds this signed, multiscale risk measure into the usual expected-return versus risk optimization. They aggregate optimal weights across many time scales and fluctuation orders so that investor horizons and the importance of large versus small moves are both taken into account. To test the idea, they use controlled synthetic data — two-component ARFIMA (auto-regressive fractionally integrated moving average) and Markov-switching multifractal processes — which let them inject prescribed multiscale and multifractal dependence. They also apply the method to actual multi-asset financial data and compare it with mean–variance, mean–DCCA (detrended cross-correlation analysis), and mean–MFDCCA (multifractal detrended cross-correlation analysis) benchmarks.

At a high level MFCCA works like this. For each pair of asset return series the method creates running “profiles,” splits them into segments of length s, and removes local trends (here with a degree-2 polynomial fit). It then computes a local detrended cross-covariance in each segment. MFCCA aggregates those local values while keeping their sign: the signed local covariances are raised to a power related to q and then averaged. By contrast, MFDCCA first takes absolute values and loses the sign information before averaging. The parameter q controls whether the analysis emphasizes large fluctuations (q>0) or small ones (q<0). The authors use a recommended scaling range for exponent estimation of roughly 20 ≤ s ≤ N/5, where N is the series length, to avoid small-scale detrending artifacts and large-scale finite-size bias.